A Frame Theory of Hardy Spaces with the Quaternionic and the Clifford Algebra Settings

Abstract

The purpose of this article is twofold. The first is to construct frames of the $$L^2(\mathbb R^n)$$ by using dilation and modulation starting from a single function of a certain type. The second is to construct frames of the $$H^2(\mathbb R^n_{1,+})$$ by using a Cauchy type integral. The work is motivated by the recent development of sparse representation of Hardy space functions, and, especially, by adaptive Fourier decomposition in relation to rational orthogonal systems. We work in two contexts. One is the quaternionic space and the other is the Euclidean space in the Clifford algebra setting. We also investigate what type of functions can give rise to frames of $$L^2(\mathbb R^n)$$ by dilation and modulation.